| L(s) = 1 | + 3·3-s + 4.54·5-s + 9·9-s + 40.7·11-s − 53.2·13-s + 13.6·15-s − 4.54·17-s + 122.·19-s − 131.·23-s − 104.·25-s + 27·27-s − 216.·29-s − 251.·31-s + 122.·33-s + 11.8·37-s − 159.·39-s + 111.·41-s − 369.·43-s + 40.9·45-s − 262.·47-s − 13.6·51-s − 567.·53-s + 185.·55-s + 367.·57-s + 839.·59-s + 485.·61-s − 242.·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.406·5-s + 0.333·9-s + 1.11·11-s − 1.13·13-s + 0.234·15-s − 0.0649·17-s + 1.48·19-s − 1.19·23-s − 0.834·25-s + 0.192·27-s − 1.38·29-s − 1.45·31-s + 0.644·33-s + 0.0528·37-s − 0.656·39-s + 0.425·41-s − 1.30·43-s + 0.135·45-s − 0.815·47-s − 0.0374·51-s − 1.46·53-s + 0.454·55-s + 0.854·57-s + 1.85·59-s + 1.01·61-s − 0.462·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 4.54T + 125T^{2} \) |
| 11 | \( 1 - 40.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 53.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 4.54T + 4.91e3T^{2} \) |
| 19 | \( 1 - 122.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 131.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 216.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 251.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 11.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 111.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 369.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 262.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 567.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 839.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 485.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 333.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 590.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 490.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 121.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 609.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 719.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 637.T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.205758253550049286240659183678, −7.47905335322993365232125044789, −6.87022043742042983277787364342, −5.84294965280489941047744059433, −5.13849556633611255921516790337, −4.02007516798388619162543308264, −3.37897544399514458125701179119, −2.18595070889658692551586986470, −1.50019691474786066294585784667, 0,
1.50019691474786066294585784667, 2.18595070889658692551586986470, 3.37897544399514458125701179119, 4.02007516798388619162543308264, 5.13849556633611255921516790337, 5.84294965280489941047744059433, 6.87022043742042983277787364342, 7.47905335322993365232125044789, 8.205758253550049286240659183678
